On the blowup of An K along a monomial ideal

The main interest in this article is the question of determining monomial ideals such that the blowup of the a ne space Ank along these ideals is regular. We show that the blowup of the a ne space Ank along a square free monomial ideal I is regular if and only if the corresponding clutter (hypergraph) associated to I is a union of some isolated vertices and a complete d-partite d-uniform clutter. This is equivalent to saying that the facets of its Stanley-Reisner complex have mutually disjoint complements. Finally we classify all monomial ideals generated in degree at most 2 such that the blowup of the a ne space Ank along these ideals is regular.